The spontaneous emission rate for the massless and the massive case generally does not coincide. The
differences lies in the mass gap of the energy spectra which is independent of
the particle trajectory [1]}. In case of circular motion
in massless scalar field the expressions for response function was calculated
in [2]}, [3]}. Following the same analogy, the response
function for the massive scalar field is given by,
\(\mathcal {G}^{11}(E_k)&=&\mathcal {G}^{22}(E_k)\nonumber \\&=&\frac{a}{8\sqrt{3} \pi }e^{-2\,\sqrt{3}\,\frac{\Omega (E_k,m)}{a}}+\frac{\Omega (E_k,m)}{2 \pi };\qquad (E_k>m)~,\nonumber \\\nonumber \\\mathcal {G}^{12}(E_k)&=&\mathcal {G}^{21}(E_k)\nonumber \\&=& \frac{a}{4 \pi }\frac{e^{-\frac{\Omega (E_k,m)}{a} \sqrt{6 \sqrt{1+a^2 L^2/3}+6}}}{\sqrt{6 (\sqrt{1+a^2 L^2/3}+6)(1+a^2L^2/3)}}+ \frac{a}{2\pi }\frac{\sin \big (\frac{\Omega (E_k,m)}{a}\sqrt{6 \sqrt{1+a^2L^2/3}-6}\big )}{ \sqrt{6 (\sqrt{1+a^2 L^2/3}-6)(1+a^2L^2/3)}} ;\qquad (E_k>m)~. \) 
